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A176999 An encoding of the Collatz iteration of n. 3
1, 1111010, 11, 11110, 11110101, 1111011101101010, 111, 1111011101101010110, 111101, 11110111011010, 111101011, 111101110, 11110111011010101, 11110111110101010, 1111, 111101110110, 11110111011010101101, 11110111011010111010, 1111011, 1111110, 111101110110101 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
COMMENTS
Working from right to left, the sequence of 0's and 1's in a(n) encode, respectively, the sequence of 3x+1 and x/2 steps in the Collatz iteration of n. This is reverse one's complement of Garner's parity vector. Criswell mentions this encoding.
The length of a(n) is A006577(n). The number of 1's in a(n) is A006666(n). The number of 0's in a(n) is A006667(n). The number of terms having length k is A005186(k).
LINKS
Evans A. Criswell, The Collatz Problem (3x+1)
Lynn E. Garner, On heights in the Collatz 3n+1 problem, Discrete Math, 55 (1985), 57-64.
EXAMPLE
a(5)=11110 because the Collatz iteration for 5 is a 3x+1 step (0) followed by 4 x/2 steps (four 1's).
MATHEMATICA
encode[n_]:=Module[{m=n, p, lst={}}, While[m>1, p=Mod[m, 2]; AppendTo[lst, 1-p]; If[p==0, m=m/2, m=3m+1]]; FromDigits[Reverse[lst]]]; Table[encode[n], {n, 2, 26}]
CROSSREFS
Sequence in context: A235221 A060087 A229783 * A359347 A035613 A038449
KEYWORD
nonn,base
AUTHOR
T. D. Noe, Apr 30 2010
STATUS
approved

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Last modified March 29 09:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)